<h2>Problem 175</h2>
<div style="color:#666;font-size:80%;">28 December 2007</div><br />
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Define f(0)=1 and f(<var>n</var>) to be the number of ways to write <var>n</var> as a sum of powers of 2 where no power occurs more than twice. <br /><br />

For example, f(10)=5 since there are five different ways to express 10:<br />10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1<br /><br />

It can be shown that for every fraction <var>p/q</var> (<var>p</var><img src='images/symbol_gt.gif' width='10' height='10' alt='&gt;' border='0' style='vertical-align:middle;' />0, <var>q</var><img src='images/symbol_gt.gif' width='10' height='10' alt='&gt;' border='0' style='vertical-align:middle;' />0) there exists at least one integer <var>n</var> such that<br /> f(<var>n</var>)/f(<var>n</var>-1)=<var>p/q</var>.<br /><br />
For instance, the smallest <var>n</var> for which f(<var>n</var>)/f(<var>n</var>-1)=13/17 is 241.<br />
The binary expansion of 241 is 11110001.<br />
Reading this binary number from the most significant bit to the least significant bit there are 4 one's, 3 zeroes and 1 one. We shall call the string 4,3,1 the <span style="font-style: italic">Shortened Binary Expansion</span> of 241.<br /><br/>
Find the Shortened Binary Expansion of the smallest <var>n</var> for which<br /> f(<var>n</var>)/f(<var>n</var>-1)=123456789/987654321.<br /><br />
Give your answer as comma separated integers, without any whitespaces.
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